let f, g be Function; :: thesis: ( dom f = DOM S & ( for i being set st i in dom f holds
f . i = pi (S,i) ) & dom g = DOM S & ( for i being set st i in dom g holds
g . i = pi (S,i) ) implies f = g )
assume that
A5: dom f = DOM S and
A6: for i being set st i in dom f holds
f . i = pi (S,i) and
A7: dom g = DOM S and
A8: for i being set st i in dom g holds
g . i = pi (S,i) ; :: thesis: f = g
for x being object st x in dom f holds
f . x = g . x

proof

let x be object ; :: thesis: ( x in dom f implies f . x = g . x )
assume A9: x in dom f ; :: thesis: f . x = g . x
hence f . x = pi (S,x) by A6
.= g . x by A5, A7, A9, A8 ;
:: thesis: verum

end;

hence f = g by A5, A7; :: thesis: verum